AC coupling itself doesn't determine the type of booster clipping, though you'd normally only use a coupling capacitor before and after the boost circuit in an asymmetric clipping circuit that added a small a DC offset to achieve the asymmetric clipping. No point in adding components in the audio path that aren't required.

The capacitor in the audio signal path before the boost stops any DC voltage getting back upstream of the boost/clipping section, and the capacitor in the audio signal path after clipping section removes the applied DC offset so that you just have the AC component of the (now asymmetrically clipped) audio signal as the output. The 'booster' part of the boost/clipping section is the pedal's gain control, and will set the level of the signal passed to the clipping section. A higher signal level before the clipping section will give more signal clipping, so more overdrive/distortion.

Symmetrically clipped signals use the same type and number of diodes to clip the positive and negative halves of the audio signal. So the positive and negative halves of the waveform have the same basic shape. (In reality the initial audio waveforms themselves are rarely truly symmetrical, but assume you are passing a sine wave through the circuit).

Adding a small DC offset is one way of achieving an asymmetrically clipped signal. The DC offset shifts the whole audio signal up (or down), so by adding a +0.1Vdc offset, instead of both the positive and negative peaks reading say +2v and -2v, the positive peak now peaks at +2.1v and the negative peak is now at -1.9v. At this point, the audio signal waveform is the same basic shape, but its centre line now rests at +1v, not 0v. If the signal then passes to the diode clipping circuit, then the signal peaking at +2.1v will clip its half of the signal a lot more than the other half peaking at -1.9v. So the resulting waveform is no longer symmetrical. Diodes will typically start to clip the signal once it exceeds 0.7v, so the resulting signal is now clipped and peaking at ±0.7v. Passing the signal through another capacitor removes the DC offset, so the signal now peaking at +0.6v and -0.8v, with the shape of the top and bottom halves of the waveform looking very different.

As the maximum output level of the audio waveform is fixed by the diode forward voltage value, then it gets passed to a volume boost circuit, which can raise or lower the output level to your requirements (but without affecting the wave shape).

Another way of asymmetrically clipping a signal is to use different types of diode for the positive and negative halves of the signal, that have different forward voltage values, and so will clip at different starting signal levels. So one may clip at 0.65v and the other at 0.75v. There is no DC voltage introduced in this case, so there is no need to isolate the clipping section with capacitors either side of it.

Or for even more flavours, you could combine a DC offset with different diode types.

I believe you can also use different numbers of diodes in series for the positive and negative halves. You'll need a bigger signal level to do this, so the gain section will need to be beefier, but you can probably get more asymmetry this way (if you want to).

Note that I may have got some details of that wrong, as it's a very long time since I did any of this stuff in anger!